DG

Basis

Given a set of vectors we would like consider the vector subspace that they generate (span). This is the smallest subspace containing them. It consists of all possible linear combinations of vectors in the generating set.

Definition: Suppose V is a vector space over a field K. Given a subset B $\subseteq$ V,
span V = $\big_intersection$ {S : S is a subspace of V and B $\subseteq$ S}.

Theorem: u $\in$ span B <=> $\exists$ b₁,...b_n $\in$ B and r₁,...r_n $\in$ K such that u = r₁·b₁+...r_n·b_n.

Now consider the reverse problem: given a vector space V, find a set of generators. We could take all of V, but prefer to have as few generators as possible. If one of the generators is a linear combination of others (i.e. the set of generators is linearly dependent) then we can throw that generator away and still have a set of generators, but smaller.

A set of vectors B is linearly independent when each of the vectors is outside the span of the rest of them. In other words, none of the elements of B can be written as a linear combination of the others. Equivalently we can phrase this as follows: A subset B $\subseteq$ V is linearly independent means that the only way a linear combination of elements of B can equal 0 is by being trivial, i.e. by having all the coefficients be 0.

To see that these points of view are equivalent, suppose b₁ = r₂·b₂+...+r₂·b₂, move b₁ to the other side and obtain a nontrivial linear combination equal to 0. Conversely, given a nontrivial linear combination equal to 0, choose k such that r_k is not zero, divide the equation by r_k and move b_k to the other side of the equation.

Definition: A subset B $\subseteq$ V is linearly independent means that
$\forall$ b₁,...b_n $\in$ B and r₁,...r_n $\in$ K, r₁·b₁+...r_n·b_n = 0 => r₁ = ...r_n = 0.

If B generates V, then any vector in V can be written as a linear combination of elements of B. If B is linearly independent, then any such representation is essentially unique!

Indeed, given two such representations, we can make sure that the same elements of B are involved by letting some coefficients be 0. Now subtract the representations:
(r₁·b₁+...r_n·b_n)-(s₁·b₁+...s_n·b_n) = (r₁-s₁)·b₁+...(r_n-s_n)·b_n = 0, so r₁-s₁ = ... r_n-s_n=0.

Definition: If B spans V and is linearly independent, then B is called a basis (or a set of coordinate vectors) for V.

A basis can be thought of as a minimal set of generators or, equivalently, a maximal linearly independent set.

Example: The vector space Rⁿ has a canonical basis {e₁,...e_n}, where e_k consists of all zeros except a 1 in the k-th place.

Coordinate representation

If a basis B is finite, we may as well include all of the elements of B in our linear combinations, because if a given element of B does not appear in a particular combination, it can be added with coefficient 0. Furthermore, we can order B=(b₁,...b_n).

In this case we may represent a vector u with the n-tuple (r₁,...r_n) of unique coefficients from K (known as the coordinates of u) in the representation of u as a linear combination of elements of B.

In other words, a choice of a basis B with n elements is an identification of the vector space with Rⁿ, where b_k correspond to e_k.

Example: Consider E² with a choice of origin. Let (i, j) be an orthonormal ordered set, meaning that the line segments connecting i and j to the origin are perpendicular (orthogonal) and have unit (=1) length.

Then (i, j) form a basis known as cartesian coordinates. Each vector in E² is a unique linear combination of i and j, which can be obtained by orthogonal projection to the axes. The x and y axes are the linear subspaces spanned by i and j respectively. Orthogonal projection of a vector u to an axis, say the y-axis, is a point on the axis u_y·j. The scalar u_y is the y-coodinate of the vector. Similarly for the other axis. Then u = u_x·i+u_y·j.

The Euclidean plane E² with a choice of origin and cartesian coordinates is identified with R² where the basis (i, j) corresponds to (e₁, e₂). This is the source of two different common notations:
(v_x, v_y) = v_x·i+v_y·j.

Example: In E³ we have a set of three mutually perpendicular vectors (i, j, k) with k spanning the z-axis.

While any two bases for E² are equivalent, i.e. one is carried to the other by a suitable Euclidean transformation, this is not the case for E³, where there are two such equivalence classes of bases (orientations).

The canonical orientation is given by the so-called right hand rule: if you wrap the fingers of the right hand from i to j, then the thumb points in the k direction.

Exercise: Show that vector operations on E² correspond to the vector operations for R².

A vector space may have many different bases. However if it has a finite basis, then the number of elements in various bases remains the same and is known as the dimension of the space.

History:

Cartesian coordinates are named after René Descartes (Renatus Cartesius), whose book Geometry (published in 1637, banned by the Catholic church in 1664) did not mention coordinates.
The term coordinates was introduced by Gottfried Wilhelm Leibniz.
The idea of a basis and the notion of dimension were introduced by Hermann Grassmann in Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844)

Last updated: Jun 17 18:53 / Last fetched: Tue Dec 2 09:52:29 CST 2008